\[\large \frac{2s-3}{s^2-3s+2}\]The denominator can be factored, giving us,\[\large \frac{2s-3}{(s-2)(s-1)}\]
From here, let's try applying Partial Fraction Decomposition.
The factors in the denominator will break up like so,
\[\large \frac{2s-3}{(s-2)(s-1)} \qquad = \qquad \frac{A}{s-2}+\frac{B}{s-1}\]
Where \(\large A\) and \(\large B\) are unknown constants that we need to solve for.
The reason they're constants is because the unknown term you place on the top of each fraction, should be one degree lower than the denominator. Just something to maybe remember. :)
Ok let's try solving this,

Hmm, yes it is :) It's good if you understand why you're doing that though.
We'll start by multiplying through by the denominator on the left.\[\large \cancel{(s-2)(s-1)}\frac{2s-3}{\cancel{(s-2)(s-1)}} \qquad = \qquad \left(\frac{A}{s-2}+\frac{B}{s-1}\right)(s-2)(s-1)\]
And yes, after cancelling some stuff out on the right, you should get the thing you're thinking.

There are several ways to continue from here.
Since we have nice easy factors, the way I would recommend is by plugging in values for \(\large s\), (which is what I think you meant c; ).
If you plug in \(\large s=1\) you should be able to find B very easily.

Ummm, I'm a little rusty on my Leplace Transforms.
I think we just take the transform.
This step requires a bit of memorization I guess.
\[\huge \mathscr{L}[e^{at}]=\frac{1}{s-a}\]Hopefully I'm remembering that correctly.

Here's just a quick reminder of where we've gotten so far, and the proper way to break it up.
\[\large \mathscr{L}^{-1}\left[\frac{2s-3}{s^2-3s+2}\right]\qquad =\qquad \mathscr{L}^{-1}\left[\frac{1}{s-1}+\frac{1}{s-2}\right]\]
\[\large =\qquad \mathscr{L}^{-1}\left[\frac{1}{s-1}\right]+\mathscr{L}^{-1}\left[\frac{1}{s-2}\right]\]

Hmm I think you'll get two terms out of this one.
When your \(\large a\) is \(\large 1\), it should give you \(\large e^{t}\).
Which is what that first inverse leplace should be giving us.
It looks like you have the second term correctly figured out though.

Close this thread at the top, it's getting too long.
It's going to get laggy with all of the fancy latex formatted.
Open a new thread with your next question.
I'll look for it.
You can type @zepdrix in the comments somewhere if I'm taking too long, heh.
That will send me a page and I can find yer question easier.